Detect/check for separation and infinite maximum likelihood estimates in logistic regression

knitr::opts_chunk$set( collapse = TRUE, comment = "#>", fig.width = 6, fig.height = 6 )

The detectseparation package

detectseparation provides pre-fit and post-fit methods for the detection of separation and of infinite maximum likelihood estimates in binomial response generalized linear models.

The key methods are detect_separation and check_infinite_estimates and this vignettes describes their use.

Checking for infinite estimates

@heinze+schemper:2002 used a logistic regression model to analyze data from a study on endometrial cancer [see, @agresti:2015, Section 5.7 or ?endometrial for more details on the data set]. Below, we refit the model in @heinze+schemper:2002 in order to demonstrate the functionality that detectseparation provides.

library("detectseparation") data("endometrial", package = "detectseparation") endo_glm <- glm(HG ~ NV + PI + EH, family = binomial(), data = endometrial) theta_mle <- coef(endo_glm) summary(endo_glm)

The maximum likelihood (ML) estimate of the parameter for NV is actually infinite. The reported, apparently finite value is merely due to false convergence of the iterative estimation procedure. The same is true for the estimated standard error, and, hence the value r round(coef(summary(endo_glm))["NV", "z value"], 3) for the $z$-statistic cannot be trusted for inference on the size of the effect for NV.

@lesaffre+albert:1989[, Section 4] describe a procedure that can hint on the occurrence of infinite estimates. In particular, the model is successively refitted, by increasing the maximum number of allowed iteratively re-weighted least squares iterations at east step. The estimated asymptotic standard errors from each step are, then, divided to the corresponding ones from the first fit. If the sequence of ratios diverges, then the maximum likelihood estimate of the corresponding parameter is minus or plus infinity. The following code chunk applies this process to endo_glm.

(inf_check <- check_infinite_estimates(endo_glm)) plot(inf_check)

Clearly, the ratios of estimated standard errors diverge for NV.

Detecting separation

detect_separation tests for the occurrence of complete or quasi-complete separation in datasets for binomial response generalized linear models, and finds which of the parameters will have infinite maximum likelihood estimates. detect_separation relies on the linear programming methods developed in the 2017 PhD thesis by Kjell Konis [@konis:2007].

detect_separation is pre-fit method, in the sense that it does not need to estimate the model to detect separation and/or identify infinite estimates. For example

endo_sep <- glm(HG ~ NV + PI + EH, data = endometrial, family = binomial("logit"), method = "detect_separation") endo_sep

The detect_separation method reports that there is separation in the data, that the estimates for (Intercept), PI and EH are finite (coded 0), and that the estimate for NV is plus infinity. So, the actual maximum likelihood estimates are

coef(endo_glm) + coef(endo_sep)

and the estimated standard errors are

coef(summary(endo_glm))[, "Std. Error"] + abs(coef(endo_sep))

We can also use the glpk solver for solving the linear program for separation detection

update(endo_sep, solver = "glpk")

or use the implementation using lpSolveAPI directly

update(endo_sep, implementation = "lpSolveAPI")

See ?detect_separation_control for more options.

As proven in [@kosmidis+firth:2018], an estimator that is always finite, regardless whether separation occurs on not, is the reduced-bias estimator of [@firth:1993], which is implemented in the brglm2 R package.

library("brglm2") summary(update(endo_glm, method = "brglm_fit"))

Citation

If you found this vignette or detectseparation useful, please consider citing detectseparation. You can find information on how to do this by typing citation("detectseparation").

References